Absorbing boundary conditions are presented for three-dimensional time-dependent Schrödinger-type of equations as a means to reduce the cost of the quantum-mechanical calculations. The boundary condition is first derived from a semidiscrete approximation of the Schrödinger equation with the advantage that the resulting formulas are automatically compatible with the finite-difference scheme and no further discretization is needed in space. The absorbing boundary condition is expressed as a discrete Dirichlet-to-Neumann map, which can be further approximated in time by using rational approximations of the Laplace transform to enable a more efficient implementation. This approach can be applied to domains with arbitrary geometry. The stability of the zeroth-order and first-order absorbing boundary conditions is proved. We tested the boundary conditions on benchmark problems. The effectiveness is further verified by a time-dependent Hartree-Fock model with Skyrme interactions. The accuracy in terms of energy and nucleon density is examined as well.

中文翻译：

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R ^ {3}中与时间有关的Schrödinger型方程的吸收边界条件。

提出了与时间相关的三维三维Schrödinger型方程的吸收边界条件，以减少量子力学计算的成本。边界条件首先从Schrödinger方程的半离散近似推导而来，其优点是所得公式自动与有限差分方案兼容，并且在空间上无需进一步离散化。吸收边界条件表示为离散的Dirichlet-Neumann映射，可以通过使用Laplace变换的有理近似在时间上进一步近似，以实现更有效的实现。这种方法可以应用于具有任意几何形状的区域。证明了零阶和一阶吸收边界条件的稳定性。我们测试了基准问题的边界条件。有效性通过具有Skyrme交互作用的时变Hartree-Fock模型进一步验证。还检查了能量和核子密度方面的准确性。